Cut makes Conservative on edge (4)
I believe the answer is:
clip
'cut' is the definition.
(clip is a kind of cut)
'conservative on edge' is the wordplay.
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'on edge' becomes 'lip' (I've seen this before).
'c'+'lip'='CLIP'
'makes' acts as a link.
(Other definitions for clip that I've seen before include "hug" , "sort of joint" , "Device for holding papers together" , "Piece of film etc" , "Cut short; cartridge holder" .)